Problem 16 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$49 y^{4}-25$$

Step-by-Step Solution

Answer

The fully factored expression is \((7y^2+5)(7y-\sqrt{5})(7y+\sqrt{5})\)

Step 1: Identify the Squares

This step is about identifying the squares in the equation \(49y^{4} - 25\). The first term \(49y^{4}\) can be written as \((7y^2)^2\), and the second term \(25\) as \(5^2\) that gives \((7y^2)^2 - 5^2\).

Step 2: Apply the Formula for Difference of Squares

Now, we can apply the formula for difference of squares which is \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a = 7y^2\) and \(b = 5\), we apply the formula into the equation and get \((7y^2+5)(7y^2-5)\).

Step 3: Factorise further if possible

On a closer look, the term \(7y^2 - 5\) is another difference of squares where \(a = 7y\), \(b = \sqrt{5}\). So, it can be factored further to \((7y-\sqrt{5})(7y+\sqrt{5})\). Combining with previous step, the final factorised expression will be \((7y^2+5)(7y-\sqrt{5})(7y+\sqrt{5})\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a special factoring technique in algebra. This technique applies when two perfect square terms are subtracted from one another. It allows us to express the equation in a simpler, factored form. Understanding how to recognize these is crucial:

• The expression is a subtraction (hence the word "difference").
• Both terms must be perfect squares.

In mathematical terms, the difference of squares formula is given by: \[ a^2 - b^2 = (a + b)(a - b) \] Recognizing this pattern is like seeing a shortcut to simplification. For example, in our problem \(49y^4 - 25\), once identified as a difference of squares, it breaks down to \((7y^2)^2 - 5^2\). This leads directly to its factorization: \((7y^2 + 5)(7y^2 - 5)\). Recognizing such patterns makes working with polynomials much simpler.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, allowing abstract quantification of relationships:

• Variables represent unknown values and add flexibility.
• Constants are fixed numerical values.
• Operations include addition, subtraction, multiplication, and division.

In the given expression \(49y^4 - 25\), '49', '25', and '4' are constants, while 'y' is a variable. Each part of an expression serves a different purpose. Here, such an expression can be simplified using factoring techniques like the difference of squares. The original complex expression \(49y^4 - 25\) can turn into something more manageable, allowing for deeper analysis and problem-solving.

Square Numbers

Square numbers arise when a number is multiplied by itself. These are numbers like 1, 4, 9, 16, etc., representing areas of squares with sides of integer lengths. In our context, recognizing square terms is essential for employing the difference of squares technique.

• Example: \(7^2 = 49\) is a square number.
• The same goes for expressions: \((7y^2)^2 = 49y^4\).

In our specific problem, identifying 49 as \(7^2\) and 25 as \(5^2\) sets the stage for recognizing the entire expression \(49y^4 - 25\) as a difference of squares. This ability to see numbers in their squared forms is like unlocking a hidden power in algebra, enabling the peace of mind in simplification and further operations.